The Schrödinger equation is be pretty basic stuff but it is pretty cool. This is how it comes from the energy conservation law:
We start out with de Broglie (and the theory of relativity)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhA2HvtGfJ-_F9673PVvWbPK91cm3_3T6WgrWYXf68KxNE_Y2UcWwlk28P40EE0b0jRs0euXOnr_2c4n-d5GVgAmDibmm6pYpILP1uD54Ku_UDeKi0l3YK3Rq8DRwDgZ5RwCAgYk6-rG-A/s400/index_1.gif)
and we know that all particles are also waves, a typical wave equation being
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDqTHzB34gjzb-iGdTkfZPLU-g0PMdMU2u2vrTPVvjrP3ZjQHU0OB8jn6LdJtAejs1iZJ5e4Gzaoj48cFnxTbxRoDGe71-JXZALKwRI0fu3ctmVu94BFEjriTKMd9m0T9uJHnOthLZ95Y/s400/index_2.gif)
Just for the fun of it we make the derivative
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxVuL7wgNfxbwGOJpgWKsb5KlN0BaB7hgUJWjTwBFpZkks3qVYOG80vD3WlKWWLhb8qpUobpM_1MaSeGnlE-fQLViobfhpXElLBF1pkBWq3RcrdLD-pdUxNkFbo9LArVXYis_7fxnAPCk/s400/index_3.gif)
and we find out that we can get the momentum through deriving the function. Who would have thought that?
Next we define the momentum operator
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7xQ_KAHgtGUQVTHS4RZ6p-ExMG0ZsUqkH28k6t1R0NkvFGMa0tKQxlTSaXf5IC3ZyANcAFVi-1Zpu9xua1tBbAE3MVdyK_tT2f_lBG4ih0bvTFvRnjusZJX0eTcetYRevOWVz2w6110k/s400/index_4.gif)
and we assume that for every possible wave function the following is true (this and the definition of the position operator are the axioms of quantum mechanics):
With this knowledge we can look at the energy conservation law of mechanics (T ... kinetic energy, V ... potential energy, E ... total energy)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-W7Rz7mo9XlM2zsX50xOSQXaZZeH14zb883HTDtQeRTj2rUiclobO_4mVTX1QNpnb4P2ltSlaYm95H5-tIp6ixOn5KrsirVq3wPgfBesV0fJG3ZK_blE6NfpdRbFndfswMnoY9npkzoA/s400/index_6.gif)
The kinetic energy can be rewritten
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrJLYVPgCxhtjFMBSUw7tfL9i_PLv2KWi2jGaemeWnaPHM1xRal3XKO40yeQ7nM3liuVVKAbrPECxIB9oWjbxzYuA2NwNLHjKOwkcD5UqT_uiQR9ItW8-xyG7TZ5xTAXmd1yJeNjZDXMo/s400/index_7.gif)
We can multiply this equation by Ψ
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjeE-5l7HYdDZ_IDU3YK2IruH61JTKS_JnA8FwfdQ2XmorPI0UAbGmFsHbhKtw2OrY3XjddLimKkrrPh0mU2rsqo-SFtlhg7YTI_idLfeSggqd7HwCycTZ2zFWBU9p_PnFGqtJREztF8k/s400/index_8.gif)
Finally we replace the momentum by the momentum operator
3 comments:
Isn't the momentum operator [h/(2πi)](d/dx) = -ihbar(d/dx)
thanks for the correction, I fixed it. I don't know why I moved the 2π from the numerator into the denominator in the third equation.
Schrödinger equation is blogged!
That's pretty cool!
Recently I've made a QM post too:
Are Orbitals Real?!
Would appreciate your opinion on that.
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